Tool changing systems for program-controlled universal milling machines and machining centers, as they are called, are already known in various configurations. Normally, these comprise one or several grippers by means of which the tools are removed from the magazine, which is typically disposed in the rear portion of the machine, transported in front of the respective work spindle and automatically loaded therein. The dimensions and the arrangement of the tool magazine relative to the two spindle axes requires complex three-dimensional movements of the grippers and of their supporting structure composed of a multiplicity of individual components, which movements can only be performed by a correspondingly large number of discrete drive mechanisms. This applies especially to tool changing devices of the type in which the horizontal and the vertical work spindles are selectively loaded. The relatively large number of separately movable components in such devices renders the control complex and results, in addition to high manufacturing costs, in a considerable space requirement in the immediate vicinity of the machine tool. Moreover, reliability and tool changing times are usually unsatisfactory with such complex changing systems.